A simple propositional calculus for compact Hausdor spaces (original) (raw)
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A strict implication calculus for compact Hausdorff spaces
Annals of Pure and Applied Logic, 2019
We introduce a simple modal calculus for compact Hausdorff spaces. The language of our system extends that of propositional logic with a strict implication connective, which, as shown in earlier work, algebraically corresponds to the notion of a subordination on Boolean algebras. Our base system is a strict implication calculus SIC, to which we associate a variety SIA of strict implication algebras. We also study the symmetric strict implication calculus S 2 IC, which is an extension of SIC, and prove that S 2 IC is strongly sound and complete with respect to de Vries algebras. By de Vries duality, this yields completeness of S 2 IC with respect to compact Hausdorff spaces. Since some of the defining axioms of de Vries algebras are Π 2-sentences, we develop the corresponding theory of non-standard rules, which we term Π 2-rules. We study the resulting inductive elementary classes of algebras, and give a general criterion of admissibility for Π 2-rules. We also compare our approach to approaches in the literature that are related to our work. 1
Model Completeness and Π2-rules: The Case of Contact Algebras
2020
We give a sufficient condition for deciding admissibility of non-standard inference rules inside a modal calculus S with the universal modality. The condition requires the existence of a model completion for the discriminator variety of algebras which are models of S. We apply the condition to the case of symmetric strict implication calculus, i.e., to the modal calculus axiomatizing contact algebras. Such an application requires a characterization of duals of morphisms which are embeddings (in the model-theoretic sense). We supply also an explicit infinite set of axioms for the class of existentially closed contact algebras. The axioms are obtained via a classification of duals of finite minimal extensions of finite contact algebras.
The mathcalLmn{\mathcal L}^m_nmathcalLmn-propositional calculus
Mathematica Bohemica, 2015
T. Almada and J. Vaz de Carvalho (2001) stated the problem to investigate if these Lukasiewicz algebras are algebras of some logic system. In this article an affirmative answer is given and the L m n-propositional calculus, denoted by ℓ m n , is introduced in terms of the binary connectives → (implication), ։ (standard implication), ∧ (conjunction), ∨ (disjunction) and the unary ones f (negation) and D i , 1 i n − 1 (generalized Moisil operators). It is proved that ℓ m n belongs to the class of standard systems of implicative extensional propositional calculi. Besides, it is shown that the definitions of L m n-algebra and ℓ m n-algebra are equivalent. Finally, the completeness theorem for ℓ m n is obtained.
A New Algebraic Version of Monteiro’s Four-Valued Propositional Calculus
Open Journal of Philosophy, 2014
In the XII Latin American Symposium on Mathematical Logic we presented a work introducing a Hilbert-style propositional calculus called four-valued Monteiro propositional calculus. This calculus, denoted by 4 , is introduced in terms of the binary connectives ⇒ (implication), → (weak implication), ∧ (conjunction) and the unary ones (negation) and ∇ (modal operator). In this paper, it is proved that 4 belongs to the class of standard systems of implicative extensional propositional calculi as defined by Rasiowa (1974). Furthermore, we show that the definitions of four-valued modal algebra and 4 -algebra are equivalent and, in addition, obtain the completeness theorem for 4 . We also introduce the notion of modal distributive lattices with implication and show that these algebras are more convenient than four-valued modal algebras for the study of four-valued Monteiro propositional calculus from an algebraic point of view. This follows from the fact that the implication → is one of its basic binary operations.
Derivational Modal Logics with the Difference Modality
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In this chapter we study modal logics of topological spaces in the combined language with the derivational modality and the difference modality. We give axiomatizations and prove completeness for the following classes: all spaces, T1-spaces, dense-in-themselves spaces, a zerodimensional dense-in-itself separable metric space, R n (n ≥ 2). We also discuss the correlation between languages with different combinations of the topological, the derivational, the universal and the difference modality in terms of definability.
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Advances in Modal Logic, 2022
De Vries duality generalizes Stone duality between Boolean algebras and Stone spaces to a duality between de Vries algebras (complete Boolean algebras equipped with a subordination relation satisfying some axioms) and compact Hausdorff spaces. This duality allows for an algebraic approach to region-based theories of space that differs from point-free topology. Building on the recent choice-free version of Stone duality developed by Bezhanishvili and Holliday, this paper establishes a choice-free duality between de Vries algebras and a category of de Vries spaces. We also investigate connections with the Vietoris functor on the category of compact Hausdorff spaces and with the category of compact regular frames in point-free topology, and we provide an alternative, choice-free topological semantics for the Symmetric Strict Implication Calculus of Bezhanishvili et al.
A Logic for Dually Hemimorphic Semi-Heyting Algebras and its Axiomatic Extensions
Bulletin of the Section of Logic
The variety \(\mathbb{DHMSH}\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety \(\mathbb{DHMSH}\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a Hilbert-style axiomatization of a new logic called "Dually hemimorphic semi-Heyting logic" (\(\mathcal{DHMSH}\), for short), as an expansion of semi-intuitionistic logic \(\mathcal{SI}\) (also called \(\mathcal{SH}\)) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the va...
On Some Syntactic Properties of the Modalized Heyting Calculus
arXiv.org, 2016
We show that the modalized Heyting calculus [2] admits a normal axiomatization. Then, we prove some syntactic properties for this calculus, including the assertoric equipollence of it and intuitionistic propositional calculus. The latter property leads to a variant of limited separation property for the modalized Heyting calculus.